Graphing Proportional Relationships: Practice

Find the constant from any column: $k = \frac{12}{3} = 4$, so $y = 4x$.

$y = 4 \times 7 = 28$

The table goes up by $12$ for every $3$ across, which is the same rate of $4$ for each $1$.

Divide liters by minutes in each case.

$\frac{45}{3} = \frac{75}{5} = 15$

The ratios match, so the hose runs at $15$ liters per minute. The units are liters on top and minutes below, in the order you divided.

First $k = \frac{18}{6} = 3$, so $y = 3x$. Now solve $3x = 33$ by dividing both sides by $3$.

$x = 33 \div 3 = 11$

Check: $3 \times 11 = 33$. Multiplying $33$ by $3$ answers the forward question, not this one.

Divide each $y$ by its $x$. For the table with $y = 5, 9, 13$:

$\frac{5}{2} = 2.5, \quad \frac{9}{4} = 2.25, \quad \frac{13}{6} \approx 2.17$

The ratios differ, so it is not proportional, even though $y$ goes up by $4$ each time. The tables with $y = 3, 6, 9$, with $y = 7, 21, 35$ and with $y = 1, 2, 2.5$ have constant ratios $1.5$, $7$ and $0.25$.

The constant is $\frac{y}{x}$, the second coordinate over the first.

$k = \frac{2}{3}$

Dividing the other way, $\frac{3}{2}$, gives the rate read backward. Check: $\frac{2}{3} \times 3 = 2$.

The equation is $y = -4x$. Test $(3, -12)$:

$-4 \times 3 = -12$

It works. For $(-3, -12)$, $-4 \times (-3) = 12$, not $-12$. The point $(3, 12)$ has the wrong sign, and $(-12, 3)$ has its coordinates swapped.

A proportional relationship gives $0$ when the input is $0$. Here a ride of $0$ miles still costs the fee:

$\text{cost at } 0 \text{ miles} = 3 \text{ dollars}$

The cost does rise by a steady $2$ dollars each mile, and the points lie on a straight line, but that line meets the y-axis at $(0, 3)$.

At $x = 1$ the two lines are at heights $5$ and $2$.

$5 > 2$

So $y = 5x$ climbs more for each step across and is steeper. Both values of $k$ are positive, so both lines rise from left to right.

The constant is flour over milk, $\frac{y}{x}$.

$k = \frac{3}{2} = 1.5$

So $y = 1.5x$: $1.5$ cups of flour for each cup of milk. The fraction $\frac{2}{3}$ is milk over flour, the rate read the other way.

The line goes through the origin, so it is $y = kx$ with

$k = \frac{-6}{4} = -1.5.$

Then $y = -1.5 \times 10 = -15$. The sign stays negative because $k$ is negative and $x$ is positive.

Skip the column with $x = 0$ when dividing, and check that it gives $y = 0$, which it does. Then

$\frac{7}{2} = \frac{17.5}{5} = 3.5.$

The ratios match, so the table is proportional with $k = 3.5$. Nothing requires $k$ or $y$ to be whole numbers.

Every point of the situation is on the line $y = 2.5x$, and each of these four points is:

$2.5 \times 1.5 = 3.75$

But $x = 1.5$ means one and a half muffins, which the bakery does not sell. The other three points use whole numbers of muffins, $0$ included.